Generalized [formula omitted]-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

Abstract

We analyze the generalizedk-variations for the solution to the wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian motion with Hurst parameter H≥12 in time and which is white in space. The k-variations are defined along filters of any order p≥1 and of any length. We show that the sequence of generalized k-variations satisfies a central limit theorem when p>H+14 and we estimate the rate of convergence for it via the Stein–Malliavin calculus. The results are applied to the estimation of the Hurst index. We construct several consistent estimators for H and analyze these estimators theoretically and numerically.